The solution of the differential equation xdy/dx = 2x^2 + 1; x ≠ 0 with y = 1 when x = 1, is Option 1: xy = _e x Option 2: y = x^3 + 2x^2 Option 3: y = x^2 - _e x Option 4: y = x^2 + _e x

Answer: y = x^2 + _e x. Solution: dy/dx = 2x+1/x, so y = x^2+_e, x, +C. Using y=1 at x=1: 1 = 1+0+C C=0. Hence y = x^2+_e, x, .

Calculus

Differential Equations

Easy

common

The solution of the differential equation $x\frac{dy}{dx} = 2x^2 + 1$ ; $x \neq 0$ with $y = 1$ when $x = 1$ , is

$xy = \log_e|x|$

$y = x^3 + 2x^2$

$y = x^2 - \log_e|x|$

$y = x^2 + \log_e|x|$

✅ Correct Option: 4

$\frac{dy}{dx} = 2x+\frac{1}{x}$ , so $y = x^2+\log_e|x|+C$ .

Using $y=1$ at $x=1$ : $1 = 1+0+C \Rightarrow C=0$ .

Hence $y = x^2+\log_e|x|$ .

2025: 16 May Shift 1

Given differential equation, (1 + y²)dx = (tan⁻¹y - x)dy, then which of the following is/are true?

(A) Integrating factor = tan⁻¹x

(B) Integrating factor = tan⁻¹y

(D) Degree = 1

Choose the correct answer from the options given below:

2026: 12 May Shift 2

The general solution of the differential equation $ydx + (x-y^3)dy = 0$ is :

2026: 22 May Shift 2

Let y(x) be the solution curve of the differential equation $x(1+y^2)dx - y(1+x^2)dy = 0$ and it passes through the point (0, 1), then

2025: 15 May Shift 2

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$ . If it passes through the point $(0,0)$ , then which of the following is/are true?

(A) it is a straight line.

(B) it is a parabola.

(D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$

Choose the correct answer from the options given below:

2025: 13 May Shift 2

The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

2022: 6 Aug Shift 2

Match List I with List II

List I	List II
A. The number of arbitrary constants in the particular solution of differential equation of order 2	I. 1
B. The number of arbitrary constants in the general solution of differential equation of order 2	II. 0
C. The integrating factor of differential equation $\frac{dy}{dx} + \frac{1}{x}y = 3, x > 0$ , is	III. 2
D. For differential equation, $x^2 \frac{dy}{dx} + x = xy, x > 0, \lim_{x \to 0^+} y(x)$ is equal to	IV. $x$

Choose the correct answer from the options given below:

The solution of the differential equation xdydx=2x2+1x\frac{dy}{dx} = 2x^2 + 1xdxdy​=2x2+1; x≠0x \neq 0x=0 with y=1y = 1y=1 when x=1x = 1x=1, is

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

The solution of the differential equation $x\frac{dy}{dx} = 2x^2 + 1$ ; $x \neq 0$ with $y = 1$ when $x = 1$ , is